AI-তৈরি নমুনা উত্তর
ধরি, I = ∫ cos x 3 + cos 2 x d x I = \int \frac{\cos x}{3 + \cos^2 x} dx I = ∫ 3 + c o s 2 x c o s x d x = ∫ cos x 3 + ( 1 − sin 2 x ) d x = \int \frac{\cos x}{3 + (1 - \sin^2 x)} dx = ∫ 3 + ( 1 − s i n 2 x ) c o s x d x = ∫ cos x 4 − sin 2 x d x = \int \frac{\cos x}{4 - \sin^2 x} dx = ∫ 4 − s i n 2 x c o s x d x
ধরি, z = sin x z = \sin x z = sin x ∴ d z = cos x d x \therefore dz = \cos x dx ∴ d z = cos x d x
মান বসিয়ে পাই,I = ∫ d z 2 2 − z 2 I = \int \frac{dz}{2^2 - z^2} I = ∫ 2 2 − z 2 d z = 1 2 ⋅ 2 ln ∣ 2 + z 2 − z ∣ + c = \frac{1}{2 \cdot 2} \ln\left|\frac{2 + z}{2 - z}\right| + c = 2 ⋅ 2 1 ln 2 − z 2 + z + c = 1 4 ln ∣ 2 + sin x 2 − sin x ∣ + c = \frac{1}{4} \ln\left|\frac{2 + \sin x}{2 - \sin x}\right| + c = 4 1 ln 2 − s i n x 2 + s i n x + c c c c
উত্তর: 1 4 ln ∣ 2 + sin x 2 − sin x ∣ + c \frac{1}{4} \ln\left|\frac{2 + \sin x}{2 - \sin x}\right| + c 4 1 ln 2 − s i n x 2 + s i n x + c